An Earth mass terran planet is orbiting a sun-like star at 2AU. I understand the limit for CO2 greenhouse effects is 1.67AU for whatever reason (giving a conventional habitable zone limit), but perhaps other gases like methane or even hydrogen could continue from that point.
At 1AU, using the formula T=(F/σ)1/4 and plugging in an absorption of 240W m^2, you get: 255.068627752. Earth's equilibrium temperature is considered to be 255K or -18.15 celsius, but the Earth's actual average surface temperature is 16 degrees celsius or so (and rising because of global warming). Earth's atmosphere and greenhouse effect has to add 34 or more degrees celsius.
At 2AU, you should recieve 1/4 the solar flux right? So 60W m^2. Plugging that into the equation gives you 180.360756351K or -92.78924364899999 degrees celsius. Let's say -93 degrees celsius. To reach 16 degrees celsius at twice the distance from the sun, the Earth would need to add 109 degrees, so it needs a greenhouse effect just over three times as strong as what it currently has.
I assume however, that the contributions of greenhouse gases are highly non-linear and you can't just triple the CO2 concentration to get triple the effect. Triple the CO2 and it makes up 0.12% of the atmosphere. I assume this wouldn't kill us, but it's clearly not enough.
If you actually calculate the amount of extra CO2 and/or other greenhouse gases you need, like methane (which is a stronger greenhouse gas to be fair), does the end result become ridiculously toxic to humans?